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Message: <blockquote data-quote="Nidarshana_k" data-source="post: 28454032" data-attributes="member: 580041">To determine the number of integers that can be substituted for x such that the expression (4x - 17)/(5x + 9) is less than an integer, we can start by examining the values of x that make the expression equal to an integer.If the expression (4x - 17)/(5x + 9) is an integer, then the numerator (4x - 17) must be divisible by the denominator (5x + 9). In other words, there must exist an integer n such that 4x - 17 = n(5x + 9).We can solve this equation for x by multiplying both sides by the common denominator, 5x + 9, and then rearranging the terms:(4x - 17)(5x + 9) = n(5x + 9)(5x + 9)Expanding the left side and the right side, we get:20x^2 + 9x - 153 = 25x^2 + 45x + 81nCombining like terms, we get:-5x^2 - 36x + 153 = 81nDividing both sides by -5, we get:x^2 + 7x - 31 = -81n/5Since x is an integer, we can see that n must also be an integer. Therefore, the possible values of n are -81, -80, -79, ..., -1, 0, 1, 2, ..., 31.For each of these values of n, there is exactly one value of x that makes the expression (4x - 17)/(5x + 9) equal to an integer. Therefore, there are 31 + 1 = 32 values of x that make the expression equal to an integer.If we consider all the integers for x, there are an infinite number of them. However, only 32 of them make the expression equal to an integer. Therefore, the number of integers that can be substituted for x such that the expression (4x - 17)/(5x + 9) is less than an integer is infinite minus 32, which is also infinite.</blockquote>

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